A system $T$ is linear when it respects superposition. If

$$T{x_1}=y_1,\qquad T{x_2}=y_2,$$

then for any scalars $a$ and $b$,

$$T{ax_1+bx_2}=ay_1+by_2.$$

Linearity combines additivity and scaling: responses to simple inputs can be added to obtain the response to their linear combination.

A system is time invariant when shifting the input only shifts the output. If

$$T{x(t)}=y(t),$$

then time invariance requires

$$T{x(t-t_0)}=y(t-t_0)$$

for every shift $t_0$. The discrete-time definition is analogous.

A system that is both linear and time invariant is called an LTI system. LTI systems are especially useful because their behavior can be reconstructed from the response to elementary signal components.